Light scattering spectroscopy of pulydimethylsiloxane-toluene gels

HAL Id: jpa-00208725

https://hal.archives-ouvertes.fr/jpa-00208725

Submitted on 1 Jan 1977

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Light scattering spectroscopy of pulydimethylsiloxane-toluene gels

J.P. Munch, P. Lemaréchal, S. Candau, J. Herz

To cite this version:

J.P. Munch, P. Lemaréchal, S. Candau, J. Herz. Light scattering spectroscopy of pulydimethylsiloxane- toluene gels. Journal de Physique, 1977, 38 (12), pp.1499-1509. �10.1051/jphys:0197700380120149900�.

�jpa-00208725�

LIGHT SCATTERING SPECTROSCOPY OF PULYDIMETHYLSILOXANE-TOLUENE GELS

J. P.

MUNCH,

^P.

LEMARÉCHAL,

^{S. CANDAU}

Laboratoire

d’Acoustique

Moléculaire

(*),

Université

Louis-Pasteur, 4,

^rue

Blaise-Pascal,

⁶⁷⁰⁷⁰

Strasbourg Cedex,

^France

and J. HERZ

Centre de Recherches sur les

Macromolécules, C.N.R.S., 6,

^rue

Boussingault,

⁶⁷⁰⁸³

Strasbourg Cedex,

^France

(Reçu

^le

1 er juillet 1977, accepté

^{le 18 août}

1977)

Résumé. ²⁰¹⁴ La fonction d’autocorrélation de la lumière diffusée a été mesurée _pour des gels ^de

polydiméthylsiloxane-toluène

formés soit par

gonflement

de réticulats permanents, soit par dissolu- tion de macromolécules linéaires à des concentrations moyennes. Dans les deux ^cas le coefficient de diffusion

coopératif

^{varie avec} la concentration en

polymère

^{selon une loi de}

puissance

^{avec un} exposant

plus

élevé que celui qui avait été obtenu

précédemment

pour des systèmes

polystyrène-

benzène. Par ailleurs, ^{il est} montré que les modules de

compression

uniaxiale n’obéissent plus ^{à des}

lois d’échelle simples ^avec la concentration pour des réseaux gonflés par ^un liquide moins bon solvant que celui dans lequel ^a été réalisée la réticulation.

Abstract. ²⁰¹⁴ The autocorrelation function of scattered

light

^has been measured for

polydimethyl-

siloxane-toluene

gels

formed either by

swelling

permanent ^{networks or} by dissolving ^{linear macro-}

molecules at moderate concentrations. In both _cases, the cooperative ^{diffusion constant} varies with concentration according ^{to a} power law with ^an exponent larger than that obtained

previously

^for

polystyrene-benzene

systems. ^{On the other} hand, it is shown that uniaxial compression ^{moduli do}

not obey simple

scaling

laws with the equilibrium concentration for networks swollen

by

^{a diluent of}

less quality than the solvent in which the

crosslinking

has been made.

Classification

Physics ^Abstracts 61.40K201362.00201366.10

1. Introduction. ^- Traditional methods for measur-

ing

the viscoelastic

properties

^of

gels generally depend

^on mechanical devices. It has been shown

recently

^that

optical mixing spectroscopy yields

valuable information

concerning

^the

hydrodynamic properties

^{of both}

permanent

swollen networks

[1-8]

and semi-dilute or concentrated

polymer

^solutions

[8- 11].

A theoretical model has been

proposed

^which

assumes that the

light

scattered from a

gel

arises from collective excitations of the network

[1, 12, 13].

From this

model,

^the correlation function of the

polarized

^scattered

light

^{for a}

longitudinal

fluctuation of wavevector K is

predicted

^to have the form of

an

exponential decay.

^The

decay

^{rate is}

given by

T ⁼

De ^K2

^where

Dp

^{is the}

cooperative

^diffusion

constant of the chains of the network.

On the other

hand,

^a

corpuscular

model has been

proposed by

^Mc Adam et al.

[2]

and Carlson et al.

[3,

(*) Equipe ^de Recherche Associée au C.N.R.S.

4],

^{which assumes} that each macromolecule in the

gel

^{state behaves as a}

harmonically

^bound

particle executing independent

Brownian motion about a

stationary

^{mean. The}

resulting theory predicts

^a

non-exponential intensity

autocorrelation function

given

^{in terms of a} chain elastic constant and the conventional translational diffusion coefficient.

In

previous

papers, ^we have

reported

^autocorre-

lation measurements on benzene-swollen model net-

works of

polystyrene

and shown that the

hydro- dynamic

model fits the observed data. The

cooperative

diffusion constant of

gels

^{has been}

investigated

^as

a function of the method of

synthesis

of the networks and their characteristics.

In

a first

attempt,

^{we have} assumed

ideality

^of

networks,

i.e. absence of structure defects such as

pendant ^chains, cycles,

^or

entangle-

ments, ^and

interpreted

the results obtained within the framework of the rubber

elasticity theory [5-7].

From this

analysis,

^{we were led to the conclusion}

that the so-called _memory term, which relates the dimension of the elastic chain in the swollen state and the reference swollen state

respectively,

^is

strongly

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197700380120149900

dependent

^{on the}

functionality

^{of the}

cross-linking

agent.

Recently,

^{we have}

reanalyzed

^{the data}

by allowing

^the presence of

entanglements trapped

between two permanent

junction points

^{and assum-}

ing, according

^{to a}

suggestion

^{of de}

Gennes,

^that the _average distance between the cross-links was

in the first

approximation

^{identical to the}

dynamical screening length [8].

^This

assumption implies

^that

both the

cooperative

^{diffusion constant and the}

extensional modulus should

obey scaling

^{laws with}

swelling equilibrium

concentration. Such behaviour has been

effectively

^{observed for}

polystyrene-benzene gels.

Most of the networks

investigated

^were

prepared

by

^anionic

block-copolymerization

^of styrene ^with

small.

amounts of

divinylbenzene.

^{In this} type ^of

network,

each linear chain element is connected with two different branch

points

constituted

by polydivinylbenzene

nodules. The chain elements have the characteristic

sharp

^molecular

weight

distribution of

polymers prepared by

^anionic

polymerization.

A drawback of the method used here is that the actual

functionality

of the crosslinks is unknown.

In the present paper ^we report measurements of the

cooperative

^{diffusion constant of a}

homologous

series of

polydimethylsiloxane (PDMS)

^networks

swollen in toluene. _{In many} respects ^these

gels

^are

very different from the

polystyrene

networks investi-

gated previously :

- Both the _average molecular

weight

of the elastic chains and the

functionality

^{of the}

permanent

^cross- links are known

parameters.

- Branch

points

^{are formed}

by single pluri-

functional molecules instead of nodules of

appreciable

dimensions as was the case for the branch

points

^of

the

polystyrene

networks described above.

- The elastic

properties

of PDMS networks are

very different from those of

polystyrene

^networks

since the

glass

transition of PDMS occurs at a tempe-

rature far below room

temperature ( - 120 °C).

- In all

synthesis

^{the precursor}

polymer

^concen-

tration

during

network formation was much

higher

than the

equilibrium

concentration of the swollen networks. Thus the networks do not contain _any macropores.

- The

swelling degree

of the PDMS networks in toluene is

considerably

lower than that of

polysty-

rene networks swollen at

equilibrium

^{in benzene.}

2. Theoretical. ^- The essential feature ^of a

gel

^is

that each macromolecule linked to the network

by

both chain-ends is no

longer

^{free to diffuse}

through

the whole network structure but is confined to a

region

of the network

compatible

with the number of effective crosslinks. Two different

approaches

have been used to described the viscoelastic behaviour of

gels.

Mc

Adam, et

^al.

[2], Carlson, et

^al.

[3, 4]

^have

proposed

^a

simple

^{model of}

scattering

^molecules

in the

gel

^{state which assumes} each molecule to be

harmonically

^{bound and}

executing

brownian motion around a

stationary

^mean

position. Upon application

of the Ornstein and Uhlenbeck distribution function for such a

particle [14],

Carlson and Fraser

[3]

^have

derived an

expression

for the non-normalized

optical

field autocorrelation function

G(l)(T) :

^.

where I is the _average

intensity

^{of the}

field,

^{D is the}

translational diffusion constant of the

particle,

^K

is the

scattering

wavevector and _y

(sec-l)

^{the ratio}

of the chain elastic constant k

(dyne/cm)

^{to the fric-}

tional constant

f (dyne s/cm).

From

equation (1)

it follows that the line

profiles

in a

self-beating experiment

^{will not be}

simple

^lorent-

zian with a linear

relationship

between line-width and

K2. Furthermore,

the initial

amplitude

^{of the}

normalized

intensity

autocorrelation function

g(2)(O)

will have a

K-dependent

value which is

always

^less

than

2,

^{in contrast to the}

freely diffusing particle

whose

g(2)(O)

^{= 2. In} the derivation of

equation (1),

uncorrelated

Rayleigh scattering

^{has been}

assumed, neglecting

^the presence of a

large component

^of static

scattering by

^the

gels

which is due to micro-

scopic inhomogeneities.

^{Then in a later} paper

[4]

Wun and Carlson have included in the

intensity

an additional term to account for the additional static

scattering

which contributes to the

lowering

of the initial

amplitude

^of

g(2)( -r).

In the second

model, developed by Tanaka, et

al.

[1 ],

^{and de Gennes}

[12],

^the

gel

^is considered as a

continuum and the

light scattering

is assumed to arise from

longitudinal

deformation modes of the network. The deformation of the swollen network has been shown to

obey

^{a diffusion}

equation

^and

the correlation function of the

polarized

^scattered

light

^is

given by :

The

cooperative

^{diffusion constant}

De

of the chains of the network is

given by :

1"B il)’B

wnere p (clyne/cm-) ^{is tne} longitudinal compressionai modulus and 0

(dyne s/cm4)

^the frictional force per unit volume of the network as it moves with unit

velocity

^{relative to the}

surrounding liquid.

p and 0 are

given respectively by [ 12] :

- , ,

vo

being

^number

density

of network chains in the swollen

state, 17

^the

viscosity

^{of the}

swelling liquid

and

Rh hydrodynamic

^{radius of one chain.}

By combining equations (3), (4)

^and

(5)

^{one obtains :}

where

Df

stands for the translational diffusion coefficient for a dilute solution of free macromolecules

having

^the

hydrodynamic

^radius

Rh.

This

hydrodynamic theory

^{leads to} results which

are

quite

different from those derived from the har-

monically

^bound

particle ^model,

^{since it}

predicts

an

exponential

^time

decay

^{of the} autocorrelation function of the

optical

field with a linear

relationship

between

decay

^{rate and}

^K2.

Uncorrelated

Rayleigh scattering

^has been assumed so that no account has to be taken of the static

scattering

^{due to}

spatial

non-randomness of the

crosslinking.

This additional

component

^{acts as a local}

heterodyning

^source.

Therefore,

^if

Io

stands for the

intensity

of the static

component (including

^dust

trapped

^{in the}

network)

and

7g

^{for the}

intensity

^{scattered from}

longitudinal fluctuations,

the normalized

intensity

autocorrelation function is

given by [15] :

When

Io > I,,, equation (7)

^{reduces to :}

In

previous

papers, ^we have shown that

g2(z) obeys equation (8)

^for

polystyrene-benzene gels [7].

According

^to

equation (6),

^the

cooperative

^diffusion

constant

depends

^on the dimension of the elastic chains

joining

^two crosslinks which in tum

depends

on the

swelling equilibrium

concentration

Ce.

^The

Flory theory

^of

swelling predicts

^the

following dependence

^of

Ce

^on the molecular

weight Me

^of

the elastic chain

[16]

However,

this result has been derived from ^an _expres- sion of the free _energy which has been shown to be incorrect and which leads to correct results

only

in dilute solutions

[17].

A new

approach,

^{based on}

scaling

law theories has been

recently proposed, describing

static and

dynamical properties

of both dilute and semi-dilute solutions

[13, 17, 18].

Semi-dilute solutions can be considered as net-, ^t works with a finite lifetime. The _average distance between

neighbouring

crosslinks is

given by

^the

screening length ç

^which

depends only

^{on the concen-}

tration, according

^{to the}

following scaling

^law

[13]

Therefore,

^{the diffusion constant}

D,

^is

given by

In swollen networks at the

equilibrium

state, the _average distance between

neighbouring

^cross-

links

depends

^{on the}

swelling equilibrium

^concen-

tration

Ce.

^{De Gennes}

suggested

that this distance is

given

^{as in the case of}

interpenetrated

^solutions

by

the

screening length j.

^This

assumption implies

^{that :}

The

equilibrium

concentration is

given

in the first

approximation by :

where

Me represents

^{the molecular}

weight

^{of the}

elastically

effective chains

connecting

^two

junction points regardless

^{of their} nature,

entanglements

^or

permanent crosslinks. For ideal networks

prepared

from a precursor

polymer

of known molecular

weight M p’ ç

^is

^equal

^to the radius of

gyration RF

of the _precursor

polymer.

Therefore :

where C* is the cross-over concentration between dilute and semi-dilute

regions

^{for a} solution of macromolecules of molecular

weight Mp.

Combination of

equations (12)

^and

(13)

^leads

to the

following

concentration

dependence

^of

Dc

Then,

^{one can}

predict

^{the same}

exponent

^{for both}

scaling

^laws

D,,

⁼

f (C )

^and

De

⁼

f (Ce)

^{relative to}

semi-dilute solutions and swollen

networks,

^res-

pectively.

This result has been verified

experimentally

in

polystyrene gels [8].

It is also

interesting

^to consider the concentration

dependence

^{of the}

compressional

^{modulus E. In}

Flory’s theory, E

^is

proportional

^to the number

density

^of

elastically

effective chains in the

dry

state

[16].

^{As a}

consequence, E

^oc

Ce2.

On the other

hand, scaling

^law

theory predicts

that E is

proportional

^to the number

density

^of

elastic chains in the swollen state,

resulting

^{in the}

following scaling

^{law for E}

[13]

Such a

dependence

^{of E on}

Ce

has been observed in

polystyrene

networks swollen

by

^benzene.

3.

Expérimental.

^{- 3.1} PREPARATION AND CHA- RACTERISTICS OF SAMPLES. ^-

Polydimethylsiloxane

networks were obtained

by

the addition reaction of

(a - co) dihydropolydimethylsiloxane

precursor

polymers

^with

plurifunctional allyloxy compounds, using H2PtC’,,

^{as a}

catalyst.

The method has been described earlier

[19].

^These

crosslinking

^reactions

were carried out in the _presence of toluene at 60 OC.

Triallyloxy-1,2,3

propane,

tetraallyloxyethane,

^and

bis

allyloxy-3 dimethylallyloxy-2,2

propane oxide

[20]

were used as

3, 4

and 6 functional

crosslinking

agents.

The _precursor

polymers

^{were chosen in a molecular}

weight

range between 4 500 and 17 000.

The volume

swelling degree

of the networks at

equilibrium

^has been determined with an accuracy of about

5 %, using

^a

procedure already

^described

^[21].

The

experimental technique

^{and the}

apparatus [22, 23]

used for unidirectional

compression

measurements have also been described elsewhere

[24].

^All compres- sion measurements were carried out on networks swollen at

equilibrium

in toluene at small deformation ratios 0.8 A 1.

In table 1 are listed the

network-samples

^{and their}

characteristics.

Table II shows the molecular

weights Mn

^{of the}

linear PDMS

samples

^{used in our}

experiments,

determined

by

^chemical

endgroup analysis. Sample

¹

is a linear PDMS obtained

by

^{an anionic}

polyme-

rization method

(1). Samples

^{2-5 are}

(a - co) hydro- genosilane polydimethylsiloxanes (2),

which have been used for the

preparation

of networks.

TABLE 1

PDMS swollen in toluene

(a)

e) The networks have been prepared in toluene at 70 °C at a

concentration of 83 % .

(b) ^Number average molecular weight of the precursor polymer

determined by endgroup analysis.

(C) Functionality ^{of the} crosslinking agent.

(a) Data of Belkebir Mrani et al. [24]. (E is related to the para- meter G * of the authors through ^the relationship ^{E = G *}

q¡Õ 1/3,

where _qio is the swelling equilibrium ratio.)

(e) Determined from light scattering spectroscopy.

(1) The ^{authors are} grateful ^to Dr. S. Boileau who has prepared

this high ^molecular weight polymer.

(2) ^These polymers ^were prepared by the Silicon Division of Rhône-Poulenc.

TABLE II Linear PDMS

(1) ^The weight average molecular weight, the radius of gyration

and the second virial coefficient in toluene, have been determined

by conventional light scattering.

We have no exact information

concerning

^the

polydispersity

^{of the}

samples. However,

^an

analysis

of the autocorrelation function of the

photocurrent, performed by

the method of cumulants

[25, 26, 15]

showed that the

polydispersity

^index

(ratio

^of

weight

average molecular

weight

^to the number _average molecular

weight)

^{does not} exceed 1.3. Such a value of

Mw/Mn

^{is not}

negligible,

^{but this}

point

^{is not of}

great importance

^{for the} purposes of our

study.

3. 2 LIGHT SPECTROSCOPY. - The spectrometer ^and autocorrelator for

intensity

autocorrelation measure- ments have been described before

[15].

^The

light

source was an argon ion laser

(Spectra Physics

Model

165)

^{with a}

wavelength

^{of 488 nm. The cubic}

shaped gel samples (-

¹

cm3)

^were

put

into standard

glass

^cells

containing

^{an excess of}

swelling liquid.

The scattered

light

^{was collected at a}

pre-determined angle by

^{a lens} aperture

system

and focused onto the sensitive part ^{of a}

photomultiplier (ITT

^FW

130)

cathode. The

photocurrent

^was

analyzed

^after

passing through

^an

amplifier-discriminator by

^{a 24-channel}

digital

autocorrelator

(Precision

Devices and

Systems,

Ltd Malvem

system 4300).

^{The data from the corre-}

lator were

analyzed by

the method of cumulants

[25, 26, 15]

which allows for a distribution of

decay

rates

G (F)

^and

makes

^it

possible

^to calculate the average

decay rate F

and the second-order norma-

lized moment

(Jl2/f2)

^{about the mean of f.}

We have mentioned in the theoretical section that the

intensity

scattered from

longitudinal

^fluctua-

tions of swollen networks must be

heterodyned

^to

some extent

by

the static

component

^{due to micro-}

scopic heterogeneities. Furthermore,

^no

special

^care

has been taken to eliminate dust in the

reagents

^and the solvent used for the network

synthesis ;

^{the dust}

trapped

^{in the}

gel

will also contribute to the

heterodyn- ing.

^{As a} consequence, the initial

amplitude 1 gl2@(0) 1

of the normalized

photocount clipped

correlation function must be lower than that obtained for

polymer

solutions in an

homodyne self-beating experiment,

which in

practice

is about 1.75 on account of incom-

plete spatial

^coherence.

As a matter of

fact,

^it turns out that for all the

investigated gels, gk2(O) 1

⁼ 1.01-1.1. These _very

small values

of 1 gL2)(O) indicate

^{that the}

^heterodyne

component

^{dominates the}

homodyne component

and hence

equation (8)

^must

apply.

In order to check this last

point

^we have measured the autocorrelation function obtained

by mixing

the scattered

signal

^{with an external}

signal.

^{In the}

scattering angle

range of

60-900,

^{the initial}

amplitude

of the normalized autocorrelation function has been found to be reduced

further,

^{but the}

decay

^{time is}

not affected.

Alternatively,

^{a two} hundred-channel real time

wave

analyzer (Saicor

Model SAI 21

B)

^{was used to}

measure the spectrum ^{of the}

photocurrent

^{from the}

photomultiplier.

^{In all cases where the}

experimental

data were described

by

^a

single decay

^rate

r,

^both

wave

analyzer

and autocorrelator led to identical values of r within 2

%.

All the measurements were

performed

^{at room}

temperature (23 OC).

^The

polydimethylsiloxane

^net-

works swollen

by

^{toluene were allowed to stand}

at room

temperature

^{for at least one}

day

^{to allow the}

sample

^to stabilize in the cell. The

stability

^could

be checked

by monitoring

^the

intensity

^{of the DC}

component.

The solutions of PDMS in toluene were made dust free

by centrifugation (15

⁰⁰⁰

rev/min.).

^For

concentrations

larger

^{than 0.5 x}

^{10- 2} g. cm- 3,

^the

FIG. 1. ^- Semi-dilute solutions of linear PDMS Mw ^{= 6 x} 106 in toluene. a) c=3 x 10-2 g.cm-3 ; b) c=1.42x 10-2

g.cm-3 ;

0 A Scattered signal alone ; 1 à Scattered signal ^{mixed with}

external oscillator.

solutions of

sample

^{1 were too viscous to allow such}

a

procédure ; therefore,

^{the solutions were made}

with a

previously

clarified toluene.

However,

^some

dust remains in the solutions

giving

^{rise to a hetero-}

dyning

of the scattered

signal.

^{The rate of hetero-}

dyning

varies with both concentration and

scattering angle,

^as illustrated in

figure

^{1. This}

point

^{is of} great

practical importance

since it could lead to an incorrect

analysis

of the concentration

dependence

^{and wave-}

vector

dependence

^{of the}

decay

^rate of the correlation function. For this reason we have checked the scatter-

ing mode ^(homodyne

^or

heterodyne) by mixing

^the

scattered

signal

^{with an external}

oscillator, using

^a

Michelson

type

interferometer. In all cases where the

heterodyning

^{from dust was}

only partial,

^{we have}

measured the

decay

^rate

by using

the Michelson

geometry.

^{Table III}

gives

^the

scattering

^modes

experimentally

observed for the different concen-

trations.

4. Results. ^- 4. 1 SOLUTIONS OF PDMS IN TOLUENE.

- In

polymer

solutions of

given

concentration _c, the

shape

of the correlation function for scattered

light g(i)

^{and the K}

dependence

^{of the}

decay

^{rate r}

depend

^{on the}

following

^four parameters : ^{the radius} of

gyration RF

^{of the}

chain,

^the cross-over concen-

tration

c*,

^the

scattering

wavevector

K,

^{and the} minimum wavevector

Km;n

^at which the relaxation time _iK of a

longitudinal

^{mode of} wavevector K is

equal

^to the relaxation time

T,

^for

complete

^disen-

tanglement

^{of one} macromolecule.

FIG. 2. ^- Various regimes ^for longitudinal fluctuations of wave vector K in solutions of polymer ^{in a} good ^solvent.

TABLE III

Scattering

^modes

for

^solutions

of sample

¹

(Mw

^{= 6 x}

106)

^{in toluene}

Because of the

large

ranges of

concentration,

molecular

weight

^and

scattering

wavevector investi-

gated

^{in our}

experiments,

^we have obtained data in various

regimes

^defined

by

^the relative values of these

parameters.

^{For the}

clarity

of the discussion

we have

reproduced

ⁱⁿ

figure

^{2 the}

diagram given by

de Gennes

[13] showing

^the range of concentration and momentum transfer for the

predicted

^existence

of the different types of modes. One can summarize the

possible experimental

situations on the

following

way.

4.1.1 Dilute

regime

^{c c*. -}

a) Region II

g(i)

^is

exponential

Do

^{is the} self diffusion of one chain

b)

Region

^III

g(i)

^{is non}

exponential

r is the _average

decay

^{rate. One}

probes

^{inner modes}

of a

single

^chain.

4.1.2 Semi-dilute

regime

^c > c*. ^-

a) Region

¹

g(il

^is

exponential

De

^{is the}

cooperative

^{diffusion constant in the}

gel regime

b) Région

^II’

g(i)

^is

exponential

Dfree

^{is the diffusion constant in the}

disentangled regime

The différence between

D free

^and

De

^{amounts to}

a

change

^of

prefactors.

c) Region

^III’

g(i)

^{is non}

exponential r

oc

K3.

One

probes

inner modes within a coherence

length.

The behaviour which is of interest for a

quantitative comparaison

^between

cooperative

modes of swollen

permanent

networks and semi-dilute solutions is the concentration

dependence

^of

De

^{in the}

region

^I.

In order to determine

accurately

the concentration

dependence

^{of the diffusion constant in both}

regions

¹

and

II,

^{one must} determine the _ranges of

scattering

wavevector where the two

following

conditions are

fulfilled :

i)

the autocorrelation function

decays exponentially ; ii)

^the

decay

^{rate follows a}

^K2 depen-

dence. It is not easy ^to characterize with _accuracy the range of K values where the autocorrelation function

departs

^{from an}

exponential.

^{On the other}

hand,

^the

procedure

which consists of

detecting

^a

departure

^{from the K2}

dependence

^{of the}

decay

^rate

obtained

by fitting g(z)

^{to an}

exponential

^is

quite

sensitive.

Figure

³ shows the _ranges of K values where r varies like

K2,

for different

polymer

^concen-

trations. In these

domains,

^an

analysis using

^the

method of cumulants shows that the correlation function is well described

by

^a

single exponential,

except for the dilute solutions where a

slight

^distri-

bution of

exponentials

^{occurs. The} average

decay

rate of this distribution is about the same

(within

³

% )

as the

decay

^rate determined

by

^force

fitting

^a

single exponential and

the value of the second-order

moment (/12/r2)

is about - 0.1. Such a value of

/12/r2

^{would indicate a}

polydispersity

^index

M,,IM. - 1.3. However,

^{one must}

point

^{out that}

the measurements have been

performed

^{at low}

scattering angles (7°

⁹

15°)

where the effect of the

non-negligible

acceptance

angle

for scattered

light

may lead to an overestimation of

Mw/Mn.

The solid line

Kç

^{= 1}

(where ç

has been calculated from

D/Do

⁼

RF/ç) reported

^on

figure

³ represents the transition line from

regions

^{II and 1 to}

regions

^III

and III’

respectively.

^The

experimentally

^observed

deviation of T from the K2

dependence

^indicates

the existence of the concentration

dependent

^corre-

FIG. 3. ^- Semi-dilute solutions of linear PDMS MW=6 x ^{106 in}

toluene. Q c=0.47 x 10-4 g.cm-3 ; x c=4.5 x 10-4 g.cm-3 ; o c = 9 x 10 - 4 g. cm - 3;

+ c = 0.18 x 10-2 g.cm-3 ;

Z c ⁼ 0.36 x 10-2 g.cm-3 ; ^{0 c = 0.72 x 10-2} g.cm-3.

lation

length j(c)

and confirms

previous

conventional

light scattering

^data

[27].

In the

regions

^{III and}

III’,

the correlation function has been found to deviate

significantly

^from

single exponential

behaviour. The

decay

time obtained

by

^force

fitting

^a

single exponential depends

^{on the}

sampling

^{time. On the other}

hand,

^the average

decay

time determined from the cumulant

analysis

^is

insensitive to a

change

^{of the}

sampling

^{time. The}

second-order moment is of the order of

magnitude

of 0.2. We have not

attempted

^a

rigorous analysis

of the correlation

function, using

^{the exact}

expression given by

de Gennes and Dubois-Violette

[28].

^Such

an

analysis

^{has been}

performed by Adam, et

^al.

[10]

on the

system polystyrene-benzene.

Figure

⁴ shows the concentration

dependence

^of

both

Do

^and

Dc

obtained from the data of

figure

^3.

FIG. 4. ^- Semi-dilute solutions of linear PDMS samples in toluene.

+ Mw = 6 X 106; 0 Mn = 17 100;

^x Mn ^{= 4 500.}

The behaviour of the diffusion constant is

quite

different from that observed in other

polymeric systems

ⁱⁿ many respects :

- In the dilute

domain, Do

^is

independent

^{of the}

concentration.

- Within the

experimental

accuracy, the

exponent

of the _power law

De

⁼

f (c)

^{is that}

predicted by

^the

theory

- There is

quite

^a

sharp

transition between dilute and semi-dilute

solutions,

which defines with a

fairly good

accuracy the cross-over concentration c*. The concentration

cé p

^{at which}

^Do

⁼

^{f (c)}

^and

^Dc

⁼

^f(c)

intersect is found to be :

The cross-over concentration calculated from

(NA : Avogadro number)

^{is :}

In

figure

^{4 we} have also

plotted

^{the diffusion}

constant as a function of concentration for PDMS of small molecular

weights.

^{In the dilute} range,

Do

^varies

significantly

^{with c. In} the semi-dilute range the

experimental points

^{lie on the same curve}

De

⁼

f (c)

^{as that for the}

high

^molecular

weight sample.

These data

represent

^{the free diffusion}

constant

D free

^{in the}

disentangled regime (region II’),

since

they

^{have been} obtained for a range of K values much smaller than

Kmin

^{which can} be calculated from

[13] :

Typically

^for

c*/c

⁼

1/2

^{and M = 17 100}

One _can,

therefore,

conclude that for the system

investigated,

the diffusion constant follows the same

scaling

law with the concentration in both the

gel

^and

the

disentangled regimes

^without

change

^{of the}

numerical value of the

prefactor.

4.2 PDMS NETWORKS SWOLLEN IN TOLUENE TO THEIR EQUILIBRIUM ^{STATE. -} 4. 2.1

Shape of

^{the auto-}

correlation

function.

^- The results

reported

ⁱⁿ

previous

papers, related to the

light spectroscopy

^of

polystyrene

networks swollen

by

benzene and

ethyl-

acetate, demonstrated the

validity

^{of the}

hydro- dynamic

^model.

On the other

hand,

the results of Wun and Carl-

son on

polyacrylamide gels supported

the harmoni-

cally

^bound

particle

^model

[4].

Therefore we have

proceeded

^{to a} careful investi-

gation

of the autocorrelation function and of the K

dependence

^{of the}

decay

^rate for PDMS networks.

We have found that the autocorrelation function

can be described

by

^a

single exponential

^{with the same}

statistical _accuracy as for dilute solutions of mono-

disperse polymers.

The cumulant

analysis

^{leads to}

a value of

(P2/r2)

of about 0.02.

The

decay

^{rate varies as}

^K2

^for

scattering angles ranging

^{from 60 to 900. In this} range, the scattered

signal

^is

fully heterodyned by

^the

inhomogeneities present

^{in the}

sample,

^{as evidenced}

by experiments performed using

^an external oscillator.

4.2.2 The

cooperative diffusion

^{constant. - The}

experimental

values of

Dc

for PDMS networks are

listed in table I.

Figure

⁵ shows the variation of

Dc

as a function of the

polymer

concentration of the

gel

swollen to

equilibrium.

^{The data}

obey approximately

the same

scaling

^{law as} the semi-dilute

solutions,

with a

change

^{of the}

prefactor

value. However a

close

inspection

^of

figure

⁵ shows that the data are

FIG. 5. ^- Plot of De versus Ce for PDMS networks swollen in toluene. + f = 3 ; 0 f = 4 ; · f = 6. f ^{refers to the} functionality

of the crosslinking agent. ^The straight ^line represents De ⁼ f (c)

for semi-dilute solutions.

rather best accounted for

by fitting

the data of each series

of

networks of

given functionality separately.

One obtains then three

slightly

^shifted

straight

^lines

of

slopes ranging

^{from 0.9 to 1.}

4.2.3 The

compressional

^{modulus. - The data}

obtained

by Belkebir,

^{et al.}

[24]

^for

compressional

modulus E of PDMS networks swollen in toluene

are listed in table I.

Figure

^{6 shows a}

log-log plot

of E versus the

equilibrium

concentration. The data

can be fitted

roughly

^{with two}

slightly

^shifted

straight lines,

^{one line}

corresponding

^{to the}

f6 ^networks,

the other one to the

f3 and f4

networks. The

slope

of these

straight

^{lines is about 4.5.}

FIG. 6. ^- Compressional ^{modulus E versus} ce. + f ⁼ 3 ; 01= 4 ; tbf = 6.

5. Discussion. ^- In the

following discussion,

^we

shall summarize the main results of our

study,

^which

are

quite

different from those obtained on

polystyrene-

benzene systems in many respects, ^{and we shall}

attempt

^a

conjectural description

^{of the}

topology

of the networks which is able to account for the

experimental

observations.

a)

^For semi-dilute solutions of

high

^molecular

weight

^{PDMS in}

toluene,

^the _exponent ^{of the}

scaling

law

Dc

⁼

f(c)

^{is in}

good

agreement ^{with the}

predicted value, contrary

^to the results

previously

^obtained

in

polystyrene

systems. ^This

observation

may be related to the elastomeric nature of the PDMS

chain,

which is much more flexible than that of the

polysty-

rene. An alternative

explanation

^can be inferred from the rather _poor solvent

quality

of the toluene for

PDMS,

^{as shown}

by

the low value of the second virial coefficient

(cf.

^Table

II),

which is of the same

order of

magnitude

^{as that of}

polystyrene

ⁱⁿ

cyclo-

hexane at a

temperature

^of

approximately

^{200 above}

the theta

temperature.

^{In theta}

solvent,

the coherence

length

^is

proportional

^to the concentration. _{It may} be

possible

that the behaviour of PDMS solutions in toluene is intermediate between those of theta

regime

and

good

^solvent

regime, resulting

^{in a value of}

the critical

exponent

between 0.67 and 1. An

unequi-

vocal answer would be

given by

^a

study

^{of self-}

diffusion coefficient

Do

^{as a function} of molecular

weight

^{for a series of}

monodisperse samples

^{in dilute}

solutions.

Indeed,

if the system PDMS-toluene

obeys

the

good

^solvent

scaling

^laws

theory,

^then

Do

^should

vary

according

^to

^MO.6.

^The

exponent b

^{of the} power law

Do

^{oc M - b can} also be obtained from the _expo- nent a in the

viscosity

Mark-Houwink

equation [1]

^{oc Ma}

through

^the

relationship [16] :

^{b =}

(a

⁺

1)/3.

Values of b calculated from literature

viscosity

^data

range from 0.57

[29]

^{to 0.61}

[30]

^{for PDMS}

samples

of

high

^molecular

weights (>

²⁰

000)

in toluene.

These results

give

strong evidence that in PDMS toluene

systems

^{one observes}

good

^solvent

behaviour,

the values of critical

exponents being

^those

predicted by

^the

theory.

On the other

hand,

for PDMS of low molecular

weights (

²⁰

000)

the measured values of b _range from 0.5

[31]

^{to 0.53}

[24], indicating

poor solvent behaviour.

b)

^For swollen networks one

observes,

in the first

approximation,

^a

scaling

^{law for}

Dc

⁼

f(ce).

^This

result,

which confirms those

previously

obtained from

polystyrene gels, implies

^{that the} collective modes of networks swollen at their

equilibrium

^{state are}

also controlled

by

the correlation

length j

^associated

with the

equilibrium

concentration _Ce, There

is, however,

^a

systematic upward

^{shift of the curve}

Dc

⁼

f(ce)

^{relative to the}

networks,

^with

respect

to the

De

⁼

f(c)

^curve of the semi-dilute solutions.

This shift _may be attributed to a différence in the

equilibrium

conditions : swollen networks are studied in the presence of ^{an excess} of solvent and as a conse-

quence the chemical

potential

of the solvent within the network is the same as that of the _pure solvent.

On the other

hand,

in the semi-dilute solution the